Locally cyclic group

In group theory, a locally cyclic group is a group (G, *) in which every finitely generated subgroup is cyclic.

[edit] Some facts

Every cyclic group is locally cyclic, and every locally cyclic group is abelian.
Every finitely-generated locally cyclic group is cyclic.
Every subgroup and quotient group of a locally cyclic group is locally cyclic.
A group is locally cyclic if and only if every pair of elements in the group generates a cyclic group.
A group is locally cyclic if and only if its lattice of subgroups is distributive.
The torsion-free rank of a locally cyclic group is 0 or 1.

The additive group of rational numbers (Q, +) is locally cyclic -- any pair of rational numbers a/b and c/d is contained in the cyclic subgroup generated by 1/bd.
Let p be any prime, and let μ_p^∞ denote the set of all pth-power roots of unity in C, i.e.

$\mu_{p^{\infty}} = \left\{ \exp\left(\frac{2\pi im}{p^{k}}\right) : m,k\in\mathbb{Z}\right\}$

Then μ_p^∞ is locally cyclic but not cyclic. This is the Prüfer p-group.

The additive group of real numbers (R, +) is not locally cyclic -- the subgroup generated by 1 and π consists of all numbers of the form a + bπ. This group is isomorphic to the direct sum Z + Z, and this group is not cyclic.